Otto inherited the Duchy of Saxony and the kingship of the Germans upon his father’s death in 936. He continued his father’s work to unify all German tribes into a single kingdom and greatly expanded the king’s powers at the expense of the aristocracy. Through strategic marriages and personal appointments, Otto installed members of his own family to the kingdom’s most important duchies. This reduced the various dukes, who had previously been co-equals with the king, into royal subjects under his authority. Otto transformed the Roman Catholic Church in Germany to strengthen the royal office and subjected its clergy to his personal control.

After putting down a brief civil war among the rebellious duchies, Otto defeated the Magyars in 955, thus ending the Hungarian invasions of Europe. The victory against the pagan Magyars earned Otto the reputation as a savior of Christendom and secured his hold over the kingdom. By 961, Otto had conquered the Kingdom of Italy and extended his realm’s borders to the north, east, and south. In control of much of central and southern Europe, the patronage of Otto and his immediate successors caused a limited cultural renaissance of the arts and architecture. Following the example of Charlemagne‘s coronation as “Emperor of the Romans” in 800, Otto was crowned Emperor in 962 by Pope John XII in Rome.

Otto’s later years were marked by conflicts with the Papacy and struggles to stabilize his rule over Italy. Reigning from Rome, Otto sought to improve relations with the Byzantine Empire, which opposed his claim to emperorship and his realm’s further expansion to the south. To resolve this conflict, the Byzantine princess Theophanu married his son, Otto II, in April 972. Otto finally returned to Germany in August 972 and died of natural causes in 973. Otto II succeeded him as Emperor.”

This part made me scratch my head: “Otto called the feuding parties to his court at Magdeburg, where Eberhard was ordered to pay a fine, and his lieutenants were sentenced to carry dead dogs in public, a particularly dishonoring punishment.” Carry dead dogs in public? Seriously? The article incidentally has many examples of how ‘the game of thrones’ was played by Otto and his comtemporaries (strategic marriages, hidden agreements, banquets held where the people who showed up got massacred, …). Here’s a little more stuff from the article:

“In the early summer of 951, before his father marched across the Alps, Otto’s son Liudolf, Duke of Swabia, invaded Lombardy in northern Italy. From his stronghold in Swabia located just north of the Alps, Liudolf was in closer proximity to the Italian border than his father in Saxony. While the exact reason for Liudolf’s actions are unclear, dynastic concerns and family ties to Adelaide may have been a factor. Adelaide’s mother, Bertha of Swabia, was a daughter of Regelinda, the mother of Liudolf’s wife Ida, from her first marriage to Burchard II, Duke of Swabia. Liudolf, therefore, may have intervened in the Italian campaign at the request of Adelaide’s relatives. Additionally, Liudolf, 19 years old himself, did not view the idea of a young step-mother as in his best interests. Though Otto had named him as his successor, Liudolf feared any potential step-brother may usurp his claim to the German throne.

The purpose of Liudolf’s Italian campaign was to overthrow Berengar II and therefore render unnecessary Otto’s own expedition into Italy, and thus his marriage to Adelaide. While Liudolf was preparing his expedition, the Bavarian Duke Henry, Otto’s brother and Liudolf’s uncle, conspired against him; Swabia and Bavaria shared a long common border and the two dukes were involved in a border dispute. Henry influenced the Italian aristocrats not to join Liudolf’s campaign. When Liudolf arrived in Lombardy, he found no support and was unable to sustain his troops. His army was near destruction until Otto’s own army crossed the Alps. […] With the humiliating failure of his Italian campaign and Otto’s marriage to Adelaide, Liudolf became estranged from his father and planned a rebellion. […]

Word of the rebellion reached Otto at Ingelheim. In order to secure his position, he traveled to his stronghold at Mainz. The city was also the seat of Archbishop Frederick of Mainz, who acted as the spokesman for the rebels and offered himself as a mediator between Otto and the rebels, who quickly arrived in Mainz. Recorded details of the meeting or the negotiated treaty do not exist, but Otto soon left Mainz with a peace treaty favorable to the conspirators, most likely confirming Liudolf as heir apparent and approving Conrad’s original agreement with Berengar II, making the treaty contrary to the desires of Adelaide and Henry.

When Otto returned to Saxony, Adelaide and Henry persuaded the king to void the treaty. Convening the Imperial Diet at Fritzlar, Otto declared Liudolf and Conrad as outlaws in absentia.[25] […] Otto’s actions at the Diet prompted the people of Swabia and Franconia into civil war against their king. […] The Hungarians invaded Otto’s domain as part of the larger Hungarian invasions of Europe and ravaged much of Southern Germany during Liudolf’s civil war. Though Otto had installed the Margraves Hermann Billung and Gero on his kingdom’s northern and northeastern borders, the Principality of Hungary to the southeast was a permanent threat to German security.”

Enceladus seems to have liquid water under its icy surface. Cryovolcanoes at the south pole shoot large jets of water ice particles into space. Some of this water falls back onto the moon as “snow”, some of it adds to Saturn’s rings, and some of it reaches Saturn. The whole of Saturn’s E ring is believed to have been made from these ice particles. Because of the apparent water at or near the surface, Enceladus may be one of the best places for humans to look for extraterrestrial life.”

It’s not very big, here’s a to-scale size comparison made by NASA:

“In 2005 the Cassini spacecraft performed several close flybys of Enceladus, revealing the moon’s surface and environment in greater detail. In particular, the probe discovered a water-rich plume venting from the moon’s south polar region. This discovery, along with the presence of escaping internal heat and very few (if any) impact craters in the south polar region, shows that Enceladus is geologically active today. Moons in the extensive satellite systems of gas giants often become trapped in orbital resonances that lead to forced libration or orbital eccentricity; proximity to the planet can then lead to tidal heating of the satellite’s interior, offering a possible explanation for the activity.

Enceladus is one of only three outer Solar System bodies, with Jupiter‘s moon Io‘s sulfur volcanoes and Neptune‘s moon Triton‘s nitrogen “geysers” where active eruptions have been observed. Analysis of the outgassing suggests that it originates from a body of subsurface liquid water, which along with the unique chemistry found in the plume, has fueled speculations that Enceladus may be important in the study of astrobiology.[14] The discovery of the plume has added further weight to the argument that material released from Enceladus is the source of the E ring. […]

Because it reflects so much sunlight, the mean surface temperature at noon only reaches −198 °C (somewhat colder than other Saturnian satellites).[8] […]

Images taken by Cassini during the flyby on July 14, 2005 revealed a distinctive, tectonically deformed region surrounding Enceladus’s south pole. This area, reaching as far north as 60° south latitude, is covered in tectonic fractures and ridges.[2][40] The area has few sizable impact craters, suggesting that it is the youngest surface on Enceladus and on any of the mid-sized icy satellites; modeling of the cratering rate suggests that some regions of the south polar terrain (SPT) are possibly as young as 500,000 years, or younger.[2]”

“The Pax Mongolica (less often known as Pax Tatarica)[1] is a Latin phrase meaning “MongolPeace” coined by Western scholars to describe the stabilizing effects of the conquests of the Mongol Empire on the social, cultural, and economic life of the inhabitants of the vast Eurasian territory that the Mongols conquered in the 13th and 14th centuries. The term is used to describe the eased communication and commerce the unified administration helped to create, and the period of relative peace that followed the Mongols’ vast conquests. The term was coined in parallel to Pax Romana.

The conquests of Kublai Khan and his successors effectively connected the Eastern world with the Western world, ruling a territory from Southeast Asia to Eastern Europe. The Silk Road, connecting trade centers across Asia and Europe, came under the sole rule of the Mongol Empire. It was commonly said that “a maiden bearing a nugget of gold on her head could wander safely throughout the realm.”[2][3] The end of the Pax Mongolica was marked by political fragmentation of the Mongol Empire and the outbreak of the Black Death in Asia which spread along trade routes to much of the world.”
…

“Marco Polo (i/ˈmɑrkoʊˈpoʊloʊ/; Italian pronunciation: [ˈmarko ˈpɔːlo]; c.1254 – January 8–9, 1324)[1] was an Italian merchant traveler from the Republic of Venice[2][3] whose travels are recorded in Livres des merveilles du monde, a book which did much to introduce Europeans to Central Asia and China. He learned the mercantile trade from his father and uncle, Niccolò and Maffeo, who traveled through Asia, and apparently met Kublai Khan. In 1269, they returned to Venice to meet Marco for the first time. The three of them embarked on an epic journey to Asia, returning after 24 years to find Venice at war with Genoa; Marco was imprisoned, and dictated his stories to a cellmate. He was released in 1299, became a wealthy merchant, married and had three children. He died in 1324, and was buried in San Lorenzo.

“A meristem is the tissue in most plants containing undifferentiated cells (meristematic cells), found in zones of the plant where growth can take place.

Meristematic cells give rise to various organs of the plant and keep the plant growing. The Shoot Apical Meristem (SAM) gives rise to organs like the leaves and flowers. The cells of the apical meristems – SAM and RAM (Root Apical Meristem) – divide rapidly and are considered to be indeterminate, in that they do not possess any defined end fate. In that sense, the meristematic cells are frequently compared to the stem cells in animals, which have an analogous behavior and function.

The term meristem was first used in 1858 by Karl Wilhelm von Nägeli (1817–1891) in his book Beiträge zur Wissenschaftlichen Botanik.[1] It is derived from the Greek word merizein (μερίζειν), meaning to divide, in recognition of its inherent function.

In general, differentiated plant cells cannot divide or produce cells of a different type. Therefore, cell division in the meristem is required to provide new cells for expansion and differentiation of tissues and initiation of new organs, providing the basic structure of the plant body.”

Since then, connections between Wieferich primes and various other topics in mathematics have been discovered, including other types of numbers and primes, such as Mersenne and Fermat numbers, specific types of pseudoprimes and some types of numbers generalized from the original definition of a Wieferich prime. Over time, those connections discovered have extended to cover more properties of certain prime numbers as well as more general subjects such as number fields and the abc conjecture.

Despite a number of extensive searches, the only known Wieferich primes to date are 1093 and 3511 (sequence A001220 in OEIS).”

It’s worth having these guys in mind when you’re thinking about limits – “It has been conjectured (as for Wilson primes) that infinitely many Wieferich primes exist” – but only two (2!) such numbers are currently known, and: “It is now known, that if any other Wieferich primes exist, they must be greater than 6.7×1015.” Sometimes it takes a lot of time to get to infinity… Incidentally only 48 Mersenne primes, a different type of primes, are known and it’s also hypothesized that there’s an infinite number of those…

About me/this blog

This blog is mainly a site where I keep track of and share some of the stuff I read and learn. Only a small subset of the posts on this blog deal with economics – I have diverse interests, and as the category cloud in the sidebar below illustrates this blog contains posts about all kinds of stuff: Mathematics, physics, statistics, geology, geography, health care and medicine, psychology, evolutionary biology, genetics, history, anthropology, archaeology, chess, …

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"Happiness and its anticipation are […] proximate mechanisms that lead us to perform and repeat acts that in the environments of history, at least, would have led to greater reproductive success." (Richard D. Alexander)